I've come a little bit closer to the answer on this topic. I've turned it into a Multinomial Probability problem:

Four Probabilities:

(b-1)/(a*b)=> Counts towards the success of die A

(a-1)/(a*b)=> Counts towards the success of die B

1/(a*b)=> Counts towards the success of both dice

(a*b - a - b +1)/(a*b)=> Failure

Edit: I thought I had my numbers correct, but I had exchanged a digit of 6 for a digit of 7, which threw my calculations.